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Mirrors > Home > MPE Home > Th. List > sqsqrd | Structured version Unicode version |
Description: Square root theorem. Theorem I.35 of [Apostol] p. 29. (Contributed by Mario Carneiro, 29-May-2016.) |
Ref | Expression |
---|---|
abscld.1 |
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Ref | Expression |
---|---|
sqsqrd |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abscld.1 |
. 2
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2 | sqrth 12963 |
. 2
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3 | 1, 2 | syl 16 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1952 ax-ext 2430 ax-sep 4514 ax-nul 4522 ax-pow 4571 ax-pr 4632 ax-un 6475 ax-cnex 9442 ax-resscn 9443 ax-1cn 9444 ax-icn 9445 ax-addcl 9446 ax-addrcl 9447 ax-mulcl 9448 ax-mulrcl 9449 ax-mulcom 9450 ax-addass 9451 ax-mulass 9452 ax-distr 9453 ax-i2m1 9454 ax-1ne0 9455 ax-1rid 9456 ax-rnegex 9457 ax-rrecex 9458 ax-cnre 9459 ax-pre-lttri 9460 ax-pre-lttrn 9461 ax-pre-ltadd 9462 ax-pre-mulgt0 9463 ax-pre-sup 9464 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 966 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2264 df-mo 2265 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2601 df-ne 2646 df-nel 2647 df-ral 2800 df-rex 2801 df-reu 2802 df-rmo 2803 df-rab 2804 df-v 3073 df-sbc 3288 df-csb 3390 df-dif 3432 df-un 3434 df-in 3436 df-ss 3443 df-pss 3445 df-nul 3739 df-if 3893 df-pw 3963 df-sn 3979 df-pr 3981 df-tp 3983 df-op 3985 df-uni 4193 df-iun 4274 df-br 4394 df-opab 4452 df-mpt 4453 df-tr 4487 df-eprel 4733 df-id 4737 df-po 4742 df-so 4743 df-fr 4780 df-we 4782 df-ord 4823 df-on 4824 df-lim 4825 df-suc 4826 df-xp 4947 df-rel 4948 df-cnv 4949 df-co 4950 df-dm 4951 df-rn 4952 df-res 4953 df-ima 4954 df-iota 5482 df-fun 5521 df-fn 5522 df-f 5523 df-f1 5524 df-fo 5525 df-f1o 5526 df-fv 5527 df-riota 6154 df-ov 6196 df-oprab 6197 df-mpt2 6198 df-om 6580 df-2nd 6681 df-recs 6935 df-rdg 6969 df-er 7204 df-en 7414 df-dom 7415 df-sdom 7416 df-sup 7795 df-pnf 9524 df-mnf 9525 df-xr 9526 df-ltxr 9527 df-le 9528 df-sub 9701 df-neg 9702 df-div 10098 df-nn 10427 df-2 10484 df-3 10485 df-n0 10684 df-z 10751 df-uz 10966 df-rp 11096 df-seq 11917 df-exp 11976 df-cj 12699 df-re 12700 df-im 12701 df-sqr 12835 df-abs 12836 |
This theorem is referenced by: msqsqrd 13037 sqr00d 13038 zsqrelqelz 13947 nonsq 13948 prmreclem3 14090 nmsq 20838 ipcau2 20874 tchcphlem1 20875 tchcph 20877 minveclem3b 21040 efif1olem3 22126 efif1olem4 22127 cxpsqr 22274 loglesqr 22322 quad 22361 cubic 22370 quartlem4 22381 quart 22382 asinlem 22389 asinlem2 22390 efiatan2 22438 cosatan 22442 cosatanne0 22443 atans2 22452 chpub 22685 chtppilim 22850 rplogsumlem1 22859 dchrisum0flblem1 22883 dchrisum0flblem2 22884 dchrisum0fno1 22886 sin2h 28563 cos2h 28564 areacirclem1 28625 areacirclem5 28629 pell1234qrne0 29335 pell1234qrreccl 29336 pell1234qrmulcl 29337 pell14qrgt0 29341 pell14qrdich 29351 pell1qrgaplem 29355 pell14qrgapw 29358 pellqrex 29361 rmxyneg 29402 jm2.22 29485 |
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